Measurement and Analysis of Seasonal Variation Lecture 06

 Measurement and Analysis of Seasonal Variation 

Lecture 06

Introduction

The seasonal variation in time series data repeats more or less annually. The measurement of seasonal variations is done by isolating them from other components of a time series that influence the phenomenon or economic activity.

De-Trending

The process of eliminating the impacts of the trend component from time series data that have been seen. De-trending is mostly used to identify seasonal or cyclical patterns efficiently.

Consider the multiplicative time series model

Yt = T x C x S x I

CSI = TCSI / T * 100

In case of additive model

CSI = TCSI - T

Another name for the detrended time series is the stationary time series.

The following method is used to measure the seasonal variation. 

1. The Percentage of Annual Average Method

2. The Method f Ratio to Moving Average

3. The Method of Ratio to Least Squares

4. The method of Link Relative

The Percentage of Annual Average Method

First, the trend's effect is eliminated by calculating the annual simple averages, dividing each observation by the corresponding annual average, and expressing the result as a percentage.
In other words, "divide individual observations by their respective averages and multiply by 100."
The next step is to compute seasonal indices to average the percentages in order to eliminate cyclical and irregular variations. To do this, group these percentages by quarter or month, then use the mean or median to determine the average percentage for the quarter or month. The percentages under each quarter or month should be discarded. Adjust the 12 monthly or 4 quarterly average percentages using the adjustment factor if they are less than 100.

Correction factor = obtained grand mean / 400

Practice Question

Compute the seasonal indices for the four quarters by the annual percentage method from the following data of wholesale price. De-seasonalised 1992.

Year	Quarter 1	Quarter 2	Quarter 3	Quarter 4
1990	125	124	130	132
1991	119	121	116	98
1992	120	123	119	123

Solution:

Year	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Mean
1990	125	124	130	132	127.75
1991	119	121	116	98	113.50
1992	120	123	119	123	121.25

Now divide each quarterly value by their corresponding mean and express it as a percentage.

125 / 127.75 x 100 = 97.85

Year	Quarter 1	Quarter 2	Quarter 3	Quarter 4	Total
1990	97.85	97.06	101.76	103.33
1991	104.84	106.61	102.20	86.34
1992	98.97	101.44	98.14	101.44
Total	301.66	305.11	302.10	291.11
Mean	100.553	101.704	100.7	97.037	399.994

Hence the desired total is very close to 400. There is no need of adjustment.

The deseasonalisation of 1992,

Seasonal Variation by Ratio to Moving Average Method

The ratio-to-moving-average method is most frequently and widely used for the computation of seasonal indices. Each moving average is a measure of (  ), we divide each value by their corresponding moving average and express it as a percentage.

The following steps can be employed to compute the seasonal index by the ratio to the moving average method.

i. Compute the moving or centred moving of the data.

ii. Divide the observed data by their respective moving average and express them as percentages.

Each value of the series  is an index that measures the influence of seasonal and random components. These are called specific seasonal indices.

The next step is to remove the effect of the random variation to obtain seasonal indices. We arrange the seasonal relatives by months or quarters and find the monthly or quarterly averages, using either their mean or median. If the mean is to be used, compute the modified mean by discarding the unusually small or large seasonal relatives under each month or quarter.

Practice Question

A merchant’s sale of ordinary coal over a period was as shown below:

Year	Quarters
	1	2	3	4
1976	118	87	47	83
1977	94	73	41	68
1978	73	61	36	56

Compute the seasonal indices using the ratio to moving average method and use them to deseasonalise the 1977 values.

Solution:

Quarters	TCSI	4-quarter moving total	4-quarter centered moving total	4-quarter centered moving average TC
1976                1	118	-	-	-
2	87	355	-	-
3	47	311	666	83.250
4	83	297	608	76.000
1977                  1	94	291	588	73.500
2	73	276	567	70.875
3	41	255	531	66.375
4	68	243	498	62.250
1978                  1	73	238	481	60.125
2	61	226	464	58.000
3	36	-	-	-
4	56	-	-	-

Next, the given quarter value is divided by its respective centred moving average value and expressed as a percentage.

Seasonal Relative (SI)×100

-

-

46.456

109.201

127.890

103.000

61.770

109.237

121.414

105.172

-

-

Computational of seasonal indices

Year	Quarters				Total
	1	2	3	4
1976	-	-	46.456	109.201
1977	127.890	103.000	61.770	109.237
1978	121.414	105.172	-	-
Mean	124.652	104.086	54.113	109.219	392.07
Adjusted mean	127.1732	106.1912	55.2075	111.4281	399.9999

The deseasonalisation of 1977.

Quarter	Sale (TCSI)	Seasonal index SI	De-seasonalised  TC
I	94	127.1732
II	73	106.1912	68.744
III	41	55.2075	74.265
IV	68	111.4281	61.025

• Read More: Ratio to Trend Method